Near-horizon properties of trajectories with finite force relevant for Ba\~{n}ados-Silk-West effect

According to the Banados-SIlk-West (BSW) effect, two particles moving towards a black hole, can collide near the horizon with an unbounded energy in the center of mass frame. This requires one of particles to have fine-tuned parameters in such a way that the time component of generalized momentum is zero $X=0$. Thus the existence of such trjectories is a necessary condition for the BSW effect. However, it is insufficient since the forward-in-time condition requires $X>0$ outside the horizon. We examine this condition for different types of partricles and horizons and find configurations for which the BSW effect is possible. In doing so, we take into account a finite force of unspesified nature exerted on particles. It includes relationships between numbers characterizing the rate with which four-velocity, acceleration and metric functions change near the horizon. For some aforementioned relations, parameters of a system control the sign of $X$, in other cases they are required for $X$ to be real quantity. In the simplest case of free particles the BSW effect for the Kerr or Kerr-Newman black hole is impossible if a fine-tuned particle has a negative energy, so in this sense combination of the Penrose process and the BSW effect is forbidden.